fig:freqspec shows the well-known ratios between the discrete and analytical eigenfrequencies for a one-dimensional linear wave equation discretized with standard \(C^0\) -continuous finite elements of polynomial orders \(P\)  of 1 to 4. As this graph implies, the numerical eigenfrequencies of a standard Galerkin approximation always overpredict the analytical ones [1]}. Importantly, the low-frequency part of the spectra is accurate, and they become more accurate with increasing polynomial order [1]}, [3]}. The spectra corresponding to the higher-order elements exhibit jumps, leading to different “branches”. The first branch is called the acoustic branch, and is the most accurate. The other branches are called the optical branches [4]}. The maximum frequency, which limits the critical time step, is severely overpredicted in these optical branches and diverges with increasing polynomial order.
